Question

Decide whether $(x+1)$ is a factor of the polynomial $(x^3+x^2-x-1)$ or not.

Answer 1

Factor of the polynomial:

Factor theorem is a theorem that links factors and zeros of the polynomial.

According to the factor theorem, if $p\left(x\right)$ is a polynomial of degree greater than or equal to $1$ then, $\left(x-a\right)$ is a factor of $p\left(x\right)$, if $p\left(a\right)=0$, where $a$ is any real number.

Using the factor theorem,

$x+1=0$

So, $x=0–1$

$\Rightarrow$ $x=-1$

Let $p\left(x\right)=(x^3+x^2-x-1)$

Substituting $x=-1$ in $p\left(x\right)$,

$p(-1)=[{\left(-1\right)}^3+{\left(-1\right)}^2-\left(-1\right)-1]$

$\Rightarrow$$p(-1)=-1+1+1–1$ …….(${\left(-1\right)}^3=-1,{\left(-1\right)}^2=1$)

$\Rightarrow$$p(-1)=0$

Therefore, $(x+1)$ is a factor of the polynomial $(x^3+x^2-x-1)$.